Moduli stabilization, vacua and cosmological inflation in type IIA superstrings compactified on rigid Calabi–Yau threefolds

Moduli stabilization, vacua and cosmological inflation in type IIA superstrings compactified

on rigid Calabi–Yau threefolds

Yuki WAKIMOTO

Department of Physics,

Graduate School of Science and Engineering, Tokyo Metropolitan University

DISSERTATION submitted for the degree of

Doctor of Science

2018

Abstract

This dissertation is devoted to theoretical study of tring cosmol- ogy in a class of flux compactification of type IIA superstrings compactified on rigid Calabi–Yau manifolds, because these the- ories make exact calculations possible for both perturbative and non-perturbative quantum corrections, and can aovid the known

“no-go” theorems which forbid dS vacua or cosmic inflation. This gives a good theoretical laboratory for string cosmology. First, we review the modern cosmology, cosmic inflation, supergravity, superstrings, Calabi–Yau geometry, and moduli geometries with non-perturbative quantum corrections. Second, we derive a non- perturbative scalar potential. Then we find that the D-instanton corrections stabilize the axions in our model. Next, we study the remaining fields: a dilaton and Käler moduli. We search for existence of meta-stable dS vacua from various (three) view- points. First, we analyze the perturbative approximation that in- cludes a string loop correction. Next, we consider the physical bounds. Second, we investigate the one-modulus case in great detail. Third, we study meta-stability for any number of Käler moduli. Finally, we take into account the non-perturbative cor- rections numerically in one-modulus case. Then we check possi- bility of cosmic inflation. As a result, we conclude that our Uni- verse is not realized in type IIA superstring theories compactified on rigid Calabi–Yau manifolds with any number of Käler moduli.

This result is an extension of the known “no-go” theorems.

Contents

Contents 1

I Introduction 3

II Accelerating Universe 8

III Cosmic Inflation 10

III.1 Theoretical Aspects . . . . 10 III.2 Observational Aspects . . . . 13

IV Type IIA Theory of Supergravity 16

IV.1 Preparation: Construction of Spinors . . . . 16 IV.2 Supersymmetry . . . . 17 IV.3 Supergravity . . . . 18

V Type IIA Theory of Strings 20

V.1 Free Superstring . . . . 20 V.2 Interacting Strings . . . . 21 V.3 Branes . . . . 22

VI Calabi–Yau Compactification 23

VI.1 Definition of Calabi–Yau manifold . . . . 23 VI.2 Field Content . . . . 25 VI.3 Topological effects . . . . 28

VII Moduli Geometries 29

VII.1 𝒩 = 2 Vector Multiplets — Special Kähler Manifold . . 29 VII.2 Hypermultiplets — Quaternion Kähler Manifold . . . . 30 VII.3 Universal Hypermultiplet . . . . 32

VIII Non-perturbative Scalar Potential 34

VIII.1 Flux Parameters . . . . 34 VIII.2 Moment Maps . . . . 35 VIII.3 Universal Hypermultiplet . . . . 36

IX Stabilization of Axions 38

1

CONTENTS 2

X Search for dS vacua 40

X.1 Physical Bounds . . . . 41

X.2 Perturbative Approximation . . . . 41

X.3 Verification of Bounds . . . . 42

X.4 One-modulus Case . . . . 43

X.5 Generic Case: Stability Analysis . . . . 45

X.6 One-modulus Case with D-instanton Contributions . . 46

XI Search for Inflation 52

XII Conclusion and Discussion 56

Bibliography 59

I

Introduction

Our universe itself is dynamic; it was a paradigm shift in the early twentieth century. Until then, the absolute static space and time had been considered by people since A. Newton through E. Kant. Therefore A. Einstein was the first person who criticized the Kant’s philosophy caustically and persuasively.

Surely, he mentioned as follows[Ein23]:

“I am convinced that the philosophers have had a harmful ef- fect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori. [...]

This is particularly true of our concepts of time and space, which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put then in a serviceable condition.”

This door was opened by the general theory of relativity proposed by A.Einstein.

This gives the framework of some of scale phenomena; gravity or gravitational wave for example. They are the astro-scale or the low-energy scale phenom- ena respectively. And it had been tested by their observations. In addition to them, this theory gives the largest scale perspective over the astrophysics, called cosmology. The cosmos is referred as ‘u-chu’(

) in Japanese where

‘u’(

) means whole time and ’chu’(

) means whole space. Surely, the cos- mology focus on spacetime itself rather than its contents, and considers the dynamics of our universe.

Today, physicists conclude our universe is expanding acceralationally, based on some observations in this framework. For example, type Ia supernovæ have been used. Because they take normcore style collectively, their bright- ness make measuring the distance 𝑑 from us to them possible, and their spec- tra make measuring their recession speed 𝑣 possible by their red shift rates.

It was the first cosmological parameter that the factor of their proportionality called Hubble constant 𝐻 ₀ (or current Hubble variable). Theoretically, Λ-CDM model simulates its dynamics. This model considers the ordinary matter, dark matter, and dark energy as the contents of our universe, and it is the ‘standard’

cosmological model now. The details are mentioned in the chapter II.

3

CHAPTER I. INTRODUCTION 4 Moreover, the cosmological microwave background (CMB), the oldest ob- servable phenomenon for us, implies the past cosmological inflation era. This thought gives us a picture that our universe expanded exponentially in its ear- liest era we can know. The inflation model simulates this phenomenon. Es- pecially, inflation models driven by inflaton field are mainly considered. This topic is mentioned in the chapter III. Inflaton inflation model takes bottom-up approach to the cosmological inflationary phenomenon. Significantly, infla- ton inflationary models are testable by the observation, though they are oldest and highest energy phenomena. The observational test is reduced to check the values of slow-roll parameters determined by the inflaton potential from the cosmological observables.

On the other hand, there was another paradigm shift in the early twenti- eth century; the quantum field theory (QFT). Especially, H. Yukawa’s meson theory broke the European philosophical principle in different way to A. Ein- stein. Yukawa introduced an interaction between a proton, a neutron, and a meson. And they change each other indeterminately without their invariant self-identity. Heisenberg described this impact as follows[Hei62]:

“For instance, the great scientific contribution in theoretical physics that has come from Japan since the last war may be an indication for a certain relationship between philosophical ideas in the tra- dition of the Far East and the philosophical substance of quantum theory. It may be easier to adapt oneself to the quantum-theoretical concept of reality when one has not gone thorough the naive ma- terialistic way of thinking that still prevailed in Europe in the first decades of this century.”

Nowadays, the Standard Model is the most successable model of the theory of quantum fields in particle physics. This model considers the fundamental matter and three interactions; electromagnetic, weak, and strong ones. They are drawn by the Standard Model on the canvas of physics theory. Although the Standard Model can explain many quantum phenomena, it has some de- fects. The Higgs vacuum instability is one of the defects discussed for many years due to fixing Higgs mass, 𝑀 _ℎ = 125.09 ± 0.21 GeV by [ATLAS15], in ad- dition to the absence of the gravitational interaction and dark matter (it may be a known particle, however). It is predicted that the instability is at ∼ 10 ¹¹ GeV [BKKS12, DDVEM ⁺ 12, BDG ⁺ 13]. This means that we should expect the “be- yond the Standard Model” physics which makes the Standard Model “effec- tive”. We need higher energy scale theory which contains the Standard Model.

String theory is one of the candidate for post-QFT in Planck scale physics including quantum gravity. This theory considers strings with finite length ℓ _s and gives quantum fields as its oscillation modes in from-string-to-point limit ℓ _s → 0. Therefore, QFT appears as the low-energy effective theory (LEEA) of string theory. Especially, supergravity theory coupled with matter appears as the LEEA of some types of closed superstring theory. Therefore string the- ory contains the gravitational interaction the Standard Model does not have.

There are several types of supergravity/closed superstring theories. We con-

sider so-called type IIA theory. The reason will be given in the text. Type

IIA supergravity is introduced in the chapter IV Supersymmetry. And type

IIA superstring is focused in the chapter V. String theory also considers branes

CHAPTER I. INTRODUCTION 5

energy scale theory event

∼ 10 ¹⁸ GeV quantum spacetime the birth of our universe superstring

ℓ _s → 0, 𝑔 _s → 0 supergravity inflation

⋮ supersymmetry breaking

∼ 10 ² GeV “standard model”

Table I.1: Our world-picture sliced by some energy scale. The lower theory is derived by the upper theory as the effective theory. The “standard model” is used in terms of both the particle physics and the cosmology. Our assumption is that the energy scale of the supergravity and the inflation era are same. After supersymmetry braking, our known standard model interactions and matters appear. Grand unification theory may be between the supergravity and the standard model energy scale, we does not mention here, however. Inflation energy scale can be placed at different scale in different picture.

which is higher dimensional object. The fluxes given by charged branes affect the stringy fields. It is also a part of topics in the chapter V.

The background geometry influences the model building. We choose the warped product of Calabi–Yau manifold and Minkowski spacetime as the back- ground of type IIA superstring/gravity. This setting is called Calabi–Yau com- pactification. The fields from Calabi–Yau topology are called moduli. And their action is determined by moduli geometry. Calabi–Yau geometry and its effect are explained in the chapter VI. And the details of the moduli geometry is treated in the chapter VII. In general, the existence of the fluxes breaks the Calabi–Yau geometry. It is called back reaction. However, we ignore it, and ac- cept the common strategy ([KKP05, AMTV03, Str98, GS00, CDKVP01, DTV04, BKN ⁺ 10, CSS13], for instance). The reason is mentioned below.

String theory is out of our observational technology. On the other hand, the observational technology in cosmology has been grown well. Inflation model is ‘science’ today in terms of its verifiablility. Therefore, we put two assumptions; the energy scale is between inflation era and supergravity era, and the matter coupled with supergravity has inflaton. These assumptions are at a cross point of two paradigm shifts in the last century. This idea which connects string theory and cosmology is called string cosmology. It is top-down style for inflationary models. Our world-picture is described in the table I.1.

To achieve the purpose of string cosmology, we should discuss three issues.

The first is moduli stabilization. Heavy quantum particles decay into lighter ones. And massless quantum particle does not decay. This means that our theory should not include extra massless fields. This is moduli stabilization problem. The second is obtaining dS vacua. Because dS vacua is the origin of the dark energy. It is a positive vacuum energy or a cosmological constant.

When it is negative i.e. AdS vacuum, it does not suit our universe. The third is producing inflationary potential. The essence of inflaton inflation model is its shape of potential. Observations restrict the shape. Therefore, we should get an appropriate shape of the potential from the first principles.

However, several ‘no-go’ theorems are known which forbid dS vacua and

CHAPTER I. INTRODUCTION 6 inflation[MN01, IP01, HKTT07, GRLS09]. We can avoid them by including ei- ther perturbative or non-perturbative quantum corrections, or non-geometric fluxes. We focus on the first option. In the type II superstring with Calabi–Yau compactification, its LEEA has 𝒩 = 2 local supersymmetry that can determine the moduli geometry with non-perturbative quantum corrections completely.

In our case there are two types of moduli spaces; for vector multiplets and hy- permultiplets. The former geometry has been understood since a long time ago through mirror symmetry[CDLOGP91, HKTY95]. The latter geometry was understood recently ([Ale13, AMPP15] for review) due to the twistorial meth- ods which were originally developed by R. Penrose in the context of quantum gravity. Although there is only one type of non-perturbative quantum cor- rections, called NS5-instantons, out of our consideration, we can exclude it by taking an approximation because it contributes by the order e ^−1/𝑔

in contrast to the order e ^−1/𝑔

by D-instanton contribution. We also ignore back reaction here. Then the rigid Calabi–Yau background makes non-trivial explicit calcu- lations possible.

To obtain a non-trivial scalar potential, we need the gauged supergravity which is given by gauging an isometry of pre-gauged moduli spaces. In the gauged supergravity, there is a condition called tad-pole cancellation that the fluxes must satisfy. Generically, orientifold compactification is used to achieve this condition. However, the knowledge of quantum correction is not suffi- cient in this case because it reduces supersymmetry to 𝒩 = 1. In type IIA case, the tadpole cancellation condition is satisfied automatically with our choice of fluxes without Roman’s mass. Therefore, we consider type IIA theory al- though it is phenomenologically unacceptable. Our investigation is a theoreti- cal experiment as a pre-step to reach real string cosmology. The moduli space geometry with quantum corrections is introduced in the chapter VII too.

The number of the moduli fields is counted by the two Hodge numbers ℎ ^1,1 and ℎ ^2,1 in our Calabi–Yau threefolds case. And to make exact calculation pos- sible, we consider the rigid Calabi–Yau threefolds on which ℎ ^2,1 vanishes. In this case there is only one hypermultiplet so-called universal one. With these settings, we derive the non-trivial non-perturbative exact scalar potential in the chapter VIII.

Then we find that we can stabilize axions in moduli fields due to the D- instanton contributions in the chapter IX. However, the analysis of the remain- ing fields is too complicated. Therefore, we take a perturbative approxima- tion and a numerical computation. Unfortunately, the absence of the meta- stable dS vacua is implied. We verify this fact from four viewpoints. The former three are done in the perturbative approximation. First, we consider the physical bounds. And we get the condition which makes the perturba- tive approximation possible. Second, we search for dS vacua in one-modulus case i.e. ℎ ^1,1 = 1 although the Calabi–Yau manifolds with the Hodge num- bers (ℎ ^1,1 , ℎ ^2,1 ) = (1, 0) have not been constructed yet. We simply assume their existence and with these topological numbers. Then, we generically an- alyze the meta-stability for any number of ℎ ^1,1 . Finally, we consider the non- perturbative contribution in the one-modulus case numerically. They are the discussions in the chapter X.

Then we check the possibility of the inflaton inflation of our potential. In

the chapter XI, though we can find the value of the parameters that makes

CHAPTER I. INTRODUCTION 7 the potential positive, the slow-roll parameters are not acceptable for our uni- verse. Therefore, we conclude that either the perturbative or non-perturbative quantum corrections can not generate the meta-stable dS vacua and inflation.

And our research is regarded to extend the known “no-go” theorems beyond the classical level after including both perturbative and non-perturbative (D- instanton) corrections, as the conclusion of this dissertation.

The setup, deriving non-perturbative scalar potential, and searching for dS

vacua were discussed in [AKW16]. Searching for inflation was also considered

in [WK17].

II

Accelerating Universe

Space is found when all matter is removed. However the space is a dynamic thing itself. It is an implication of the general theory of relativity. Modern cos- mology treats this dynamics. Today, it is based on Λ-CDM (cold dark matter) model. Some elements of this model are introduced in this chapter. Observa- tionally, the ratio 𝐻 ₀ of the distance 𝑑 between us and the astronomical objects to their recession speed 𝑣 is important. The first goal is re-formulating of this value in the context of the general theory of relativity.

On the largest scale, our universe can be regarded as homogeneous and isotropic. The spacetime metric which satisfies the homogeneity and isotropy is given by the Friedmann–Lemaître–Robertson–Walker metric:

d𝑠 ² = d𝑡 ² − 𝑎 ² (𝑡) [ d𝑟 ²

1 − 𝑘𝑟 ² + 𝑟 ² (d𝜃 ² + sin ² 𝜃d𝜙 ² )] . (II.1) Now the light speed 𝑐 is set to 1. And the values of 𝑘, -1, 1, or 0, indicate whether the space is opened, closed, or flat by its value. In this coordinates, the space distance is measured by the round trip of the light ray. And when the space is expanding, this distance is also expanding. On the other hand, the distance is not changed in the co-moving coordinate. The co-moving distance from the point at which light starts at 𝑡 ₀ to the point at which light arrives at 𝑡 is given by

𝜒 = ∫ ^𝑡

𝑡

d𝑡 ^′ 𝑎(𝑡 ^′ ) = ∫ ^𝑎

𝑎

d ln 𝑎

𝑎𝐻 . (II.2)

𝑎(𝑡) is called scale factor and 𝐻 = ̇ 𝑎/𝑎 is Hubble parameter. The Hubble constant is 𝐻 ₀ = 𝐻(𝑡 _current ). (𝑎𝐻) ⁻¹ is the comoving Hubble radius.

The dynamics of the scale factor is governed by the Einstein equation with cosmological constant:

𝑅 _𝜇𝜈 + 1

2 𝑔 _𝜇𝜈 𝑅 = 8𝜋𝐺 _N 𝑇 _𝜇𝜈 + Λ𝑔 _𝜇𝜈 . (II.3) We assume that matter can be regarded as a perfect fluid with isotropy and homogenity on the largest scale. In this case the energy momentum tensor is

𝑇 _𝜇𝜈 = −𝑝𝑔 _𝜇𝜈 + (𝑝 + 𝜌)𝑢 _𝜇 𝑢 _𝜈 , (II.4)

8

CHAPTER II. ACCELERATING UNIVERSE 9 Table II.1: Some cosmological parameters are fixed through the observations by Planck 2015[Planck16a].

Quantity Symbol Value

Hubble paramter 𝐻 ₀ 67.31(96)

baryon density Ω _b 0.0484(10)

cold dark matter density Ω _CDM 0.258(11)

dark energy Ω _Λ 0.692(12)

relativistic matter Ω _r remaining

where 𝑝 is the pressure, 𝜌 is the energy density and 𝑢 is the velocity vector for the fluid. 𝑢 equals to (1, 0, 0, 0) in co-moving coordinates. The Einstein equation and the perfect fluid assumption give the Friedmann equations and Raychaudhuri equation,

𝐻 ² ≡ ( 𝑎 ̇ 𝑎 )

2

= 8𝜋𝐺 _N 𝜌

3 − 𝑘

𝑎 ² + Λ

3 , (II.5)

̈

𝑎 𝑎 = Λ

3 − 4𝜋𝐺 _N

3 (𝜌 + 3𝑝) . (II.6)

Now, the critical density, a density parameter 𝜌 when 𝑘 = Λ = 0, is defined by

𝜌 _c ≡ 3𝐻 ²

8𝜋𝐺 _N . (II.7)

And cosmological density parameter is defined by

Ω _tot = 𝜌/𝜌 _c . (II.8)

Friedman equation is rewritten by using the cosmological density parameter by

𝑘/𝑎 ² = 𝐻 ² (Ω _tot − 1) . (II.9) When Ω _tot equals to 1, the space is flat i.e. 𝑘 = 0. Therefore the density 𝜌 _𝑐 is called critical.

In the Λ-CDM model, the cosmological density parameter is divided into Ω _tot = Ω ⏟⏟⏟⏟⏟ _b + Ω _cdm

Ω

+Ω _r + Ω _Λ . (II.10)

They express the baryon, cold dark matter, relativistic particles, and dark en- ergy. The first two terms Ω _m are pressureless matter.

Recent survey of these cosmological parameters was done by Planck satellite, as table II.1.

Our universe is accelerationally expanding. Therefore, dark energy con-

tributes essentially. What is its origin? One candidate is vacuum energy den-

sity of quantum particles. Current observations and cosmological inflation

predict this value is positive. Therefore, we should have a positive vacuum,

a.k.a. dS vacuum.

III

Cosmic Inflation

We can derive that the universe has finite life time according to the discussion of the last chapter. This implies that two points are not correlated when their distance is over the light speed times the universe life time. The sphere with this distance as its radius is called causal horizon or Hubble sphere. Though the two distinct areas into which the causal horizon divides have no correlation in standard cosmology, the cosmic microwave background (CMB) shows the homogeneous temperature distribution over the causal horizon. The most re- cent picture of CMB by the Planck satellite is shown in the figure III.1. This is called causality problem.

One of the suggestions to solve this problem is introducing cosmic inflation which proposes the era such that the time variation of the comoving Hubble radius is negative: _d𝑡 ^d (𝑎𝐻) ⁻¹ < 0. Introducing the Hubble slow roll parameter as follows:

𝜖 _H ≡ − 𝐻 ̇

𝐻 ² , (III.1)

we can consider the time evolution of comoving Hubble radius as d

d𝑡 (𝑎𝐻) ⁻¹ = − 1

𝑎 (1 − 𝜖 _H ) < 0 ⇔ 𝜖 _H < 1 . (III.2) Therefore, our universe was expanding quasi-exponentially in the cosmic in- flationary era. Surely, when the Hubble slow roll parameter closes to zero, 𝜖 _H ∼ 0, 𝑎(𝑡) ∼ e ^𝐻𝑡 is derived.

This suggestion describes a story as follows: in the beginning, all area is correlated till the cosmic inflation put it apart. After the cosmic inflationary era, universe is divided by the causal horizon, but the correlation between its parts still remains. This story is shown in the figure III.2.

III.1 Theoretical Aspects

To describe the cosmic inflationary era concretely, the inflaton inflation models have been proposed. These models assume a homogeneous scalar field 𝜙(𝑡) called inflaton. In this chapter we review the single inflation case.

10

CHAPTER III. COSMIC INFLATION 11

Figure III.1: The temperature fluctuation observed by the Planck satellite. The range is almost ±300 μK and the average is 2.72K [Planck16b]

The inflaton 𝜙(𝑡) is included to the Friedmann equation as the matter with the energy density and the pressure below,

𝜌 = 1

2 𝜙 + 𝑉(𝜙) , ̇ (III.3)

𝑝 = 1

2 𝜙 − 𝑉(𝜙) . ̇ (III.4)

The Friedmann equation with the inflaton and the equation of motion for 𝜙 is, 𝐻 ² = 8𝜋

3𝑀 _Pℓ ² ( 1

2 𝜙 + 𝑉) − ̇ 𝑘

𝑎 ² . (III.5)

̈ 𝜙 = − 3𝐻 ̇ 𝜙 − 𝑉 ^′ (𝜙) . (III.6) where 𝑉 ^′ (𝜙) = d𝑉(𝜙)/d𝜙. We can find that 𝜙 looks as rolling down the po- tential 𝑉(𝜙) with a ‘friction term’. This friction converts the kinetic energy of the inflation 𝜙 into the cosmic expansion.

The motion of 𝜙 is governed by its potential 𝑉(𝜙). Therefore it is useful that the ‘slow roll’ parameters are defined by the potential. Potential slow roll parameters is defined by

𝜖 ≡ 𝑀 ² _Pℓ 16𝜋 ( 𝑉 ^′

𝑉 )

2

, 𝜂 ≡ 𝑀 ² _Pℓ 8𝜋 ∣ 𝑉 ^″

𝑉 ∣ . (III.7)

The slow roll condition 𝜖, 𝜂 ≪ 1 gives the necessary condition for 𝜖 _H ≪ 1.

After the inflation, the inflaton 𝜙 leaves its trace of the dynamics for the comoving curvature perturbation

𝑔 _𝑖𝑗 = 𝑎(𝑡)e ^{2ℛ(𝑡,𝐱)} 𝛿 _𝑖𝑗 , (III.8)

CHAPTER III. COSMIC INFLATION 12

inflation

1000 10 3 1 0 1 1000

0.2 0.4 0.6 0.81.0

0.01 0.1

0.001 Hubble sphere

now

light cone

scale factor

conformal time [Gyr]

50 40 30 20 10

3 10

reheatingCMB

-10 -20 -30

-40

causal contact

Figure III.2: The horizontal axis is co-moving distance. When the cosmic ex- panding speed is slower than the light speed, the Hubble sphere takes in the marginal spaces, so it is also expanding in the co-moving coordinate. On the contrary, when the cosmic expanding speed is faster than the light speed, the marginal space escapes from the Hubble sphere, so the Hubble sphere looks contracting. Though the two places on the light cone are causally detached, they can have a contact to assume the cosmic inflationary era, drawn in the lower half place of this figure. This figure is borrowed from [BM15].

and two polarizations of the gravitational wave

𝑔 _𝑖𝑗 − 𝛿 _𝑖𝑗 = ℎ _𝑖𝑗 =

⎛ ⎜

⎜ ⎜

⎜ ⎜

⎜ ⎜

⎜

⎝

0 0 0 0

0 ℎ ₊ ℎ _× 0 0 ℎ _× −ℎ ₊ 0

0 0 0 0

⎞ ⎟

⎟ ⎟

⎟ ⎟

⎟ ⎟

⎟

⎠

, (III.9)

where the coordinate is chosen by the observational frame and the transverse- traceless gauge condition is taken. Their Fourier modes relate to the primordial spectra

𝒫 _ℛ (𝑘) = 𝑘 ³

2𝜋 ² ∣ℛ _𝑘 ∣ ² , (III.10)

𝒫 _t (𝑘) = 𝑘 ³

2𝜋 ² (∣ℎ ⁺ _𝑘 ∣ ² + ∣ℎ ^× _𝑘 ∣ ² ) , (III.11)

CHAPTER III. COSMIC INFLATION 13 and they appear in the CMB temperature anisotropy 𝐶 _ℓ = 𝐶 ^𝑠 _ℓ + 𝐶 ^𝑡 _ℓ through the CMB transfer function Δ as following

𝐶 ^𝑠 _ℓ = ∫ ^∞

0

d𝑘

𝑘 Δ ^𝑠 _ℓ (𝑘)𝒫 _ℛ (𝑘) , (III.12) 𝐶 ^𝑡 _ℓ = ∫ ^∞

0

d𝑘

𝑘 Δ ^𝑡 _ℓ (𝑘)𝒫 _𝑡 (𝑘) , (III.13) where 𝐶 _ℓ is angular power spectrum. Now, the ratio between the primordial spectra with the pivot scale 𝑘 _∗ ,

𝑟 = 𝒫 _t (𝑘 _∗ )

𝒫 _ℛ (𝑘 _∗ ) , (III.14)

is called the tensor-to-scalar ratio.

III.2 Observational Aspects

Though cosmic inflation is most far phenomenon for us in our universe, in- flaton inflation models can be restricted by observations. It is most significant property of inflaton inflationary model in modern theoretical/observational cosmology. We can extract its residue form the temperature fluctuation of CMB.

A correlation between two directions 𝑛 ⃗ and 𝑛 ⃗ ^′ is formed by 𝐶(𝜃) = ⟨ Δ𝑇 _{𝑛 ⃗}

̄ 𝑇 , Δ𝑇 _{⃗ 𝑛}

̄ 𝑇 ⟩ (III.15)

where 𝜃 = ⃗ 𝑛 ⋅ ⃗ 𝑛 ^′ and

Δ𝑇 _{𝑛 ⃗} = 𝑇 _{𝑛 ⃗} − ̄ 𝑇 . (III.16) The angular power spectrum is given as its Legendre coefficient,

𝐶 _ℓ = 2𝜋 ∫ ¹

−1 d cos 𝜃𝐶(𝜃)𝑃 _ℓ (cos 𝜃) , (III.17) where 𝑃 _ℓ is Legendre polynomial. It is a point of contact of the theory and the observations.

To restrict the inflaton inflation models observationally, we put the power law ansatz,

𝒫 _ℛ (𝑘) = 𝐴 _s ( 𝑘 𝑘 _∗ )

𝑛

−1+

ln(𝑘/𝑘

)+⋯

, (III.18)

𝒫 _t (𝑘) = 𝐴 _t ( 𝑘 𝑘 _∗ )

𝑛

+

d ln 𝑘

𝑙𝑛(𝑘/𝑘

)+⋯

, (III.19)

where the exponential coefficient 𝑛 _s or 𝑛 _t is called scalar/tensor spectral tilt and 𝐴 _s or 𝐴 _t is called scalar/tensor amplitude, respectively. The tensor-to- scalar ratio 𝑟 is fixed to −8𝑛 _t . Now, the first and (𝑛 + 1)

Hubble hierarchy parameters or Hubble flow functions,

𝜖 ₁ = − 𝐻 ̇

𝐻 , (III.20)

𝜖 _𝑛+1 = 𝜖 _𝑛 ̇

𝐻𝜖 _𝑛 , (III.21)

CHAPTER III. COSMIC INFLATION 14 connect the spectral tilts and the potential slow-roll parameters[Planck16b].

Now the first Hubble hierarchy parameter is the same to the Hubble slow roll parameter. Therefore, they are interpreted as the higher order analog of the slow roll parameter. For instance, the spectral tilts and their derivatives relate to the Hubble hierarchy parameters by

𝑛 _s − 1 = − 2𝜖 ₁ − 𝜖 ₂ − 2𝜖 ₁ ² − (2𝐶 + 3)𝜖 ₁ 𝜖 ₂ − 𝐶𝜖 ₂ 𝜖 ₃ , (III.22) d𝑛 _s /d ln 𝑘 = − 2𝜖 ₁ 𝜖 ₂ − 𝜖 ₂ 𝜖 ₃ , (III.23) 𝑛 _t = − 2𝜖 ₁ − 2𝜖 ² ₁ − 2(𝐶 + 1)𝜖 ₁ 𝜖 ₂ , (III.24)

d𝑛 _t /d ln 𝑘 = − 2𝜖 ₁ 𝜖 ₂ , (III.25)

where 𝐶 = ln 2 + 𝛾 _E − 1 and 𝛾 _E is the Euler constant. And the potential slow roll parameters relate them also by

𝜖 = 𝜖 ₁ (1 − 𝜖 ₁ /3 + 𝜖 ₂ /6) ²

(1 − 𝜖 ₁ /3) ² , (III.26)

𝜂 = 2𝜖 ₁ − 𝜖 ₂ /3 − 2𝜖 ² ₁ /3 + 5𝜖 ₁ 𝜖 ₂ /6 − 𝜖 ² ₂ /12 − 𝜖 ₂ 𝜖 ₃ /6

1 − 𝜖 ₁ /3 . (III.27)

Therefore, the tensor-to-scalar ratio 𝑟 and the scalar spectral tilt 𝑛 _s are the cosmic inflationary observables which can restrict the slow-roll parameters as the theoretical parameters of the inflaton inflation models. According to the Planck 2015 result, the observables are limited and restrict inflationary models as figure III.3.

Our next question is how inflaton is obtained. We expect the higher energy

theory which gives this, namely the string theory. More specifically, we investi-

gate the supergravity theory as a low energy effective field theory of the string

theory.

CHAPTER III. COSMIC INFLATION 15

Figure III.3: The observational limits for the tensor-to-scalar ratio 𝑟 and the

scalar spectral tilt 𝑛 _s and their restriction to inflationary models. Now 𝑁 _∗ is

e-foldings defined by ln ^𝑎(𝑡 _𝑎(𝑡)

⁾ = ∫ _𝑡 ^𝑡

d𝑡𝐻 which indicates the time length of

the cosmic inflationary era. The time of the end of the inflation 𝑡 _end is defined

by 𝜖(𝑡 _end ) = 1. The pivot scale 𝑘 _∗ is taken by 0.002Mpc ⁻¹ . 𝑘 relates to the time

parameter 𝑡 by 𝑘 = 𝑎𝐻. This figure is borrowed from [BM15].

IV

Type IIA Theory of Supergravity

Our target of this dissertation is type IIA supergravity with matter multiplets on ℝ ^1,3 × 𝔜 where 𝔜 is a Calabi–Yau 3-fold with six real dimensions. Type IIA supergravity lives in 10-dimensional spacetime, and it is derived by 𝑆 ¹ com- pactification from the 11-dimensional supergravity, a.k.a. the effective the- ory of M-theory. On the other hand, 11-dimensional supergravity on ℝ ^1,3 × 𝑇 ⁷ gives 𝐷 = 4 𝒩 = 8 supergravity. It is maximal supersymmetry of 4- dimensional supergravity. And it is known that the Calabi–Yau 3-fold on the background preserves the 1/4 supersymmetry. Therefore, type IIA su- pergravity on ℝ ^1,3 × 𝔜 has 𝒩 = 2 supersymmetry. 𝒩 = 2 supersymmetry has central charge and gives BPS bound which are related to D-branes that are charged BPS states. Calabi–Yau geometry classifies the D-brane states which give the non-perturbative quantum corrections. Moreover, 𝒩 = 2 supersym- metry makes exact calculation of them possible. Those are most important motivations of our research. In this chapter, based on this motivation, we re- view the supersymmetry algebra and the derivation of type IIA supergravity from 11-dimensional supergravity, focusing on the bosonic part.

IV.1 Preparation: Construction of Spinors

Before entering the main subject, we review the way to construct spinors in general dimensions. Mathematically, spinors are a representation of the Clif- ford algebra which is a “quantization” of the exterior algebra ⨁ ^𝑑 _𝑘=1 ∧ ^𝑘 𝑉 where 𝑉 is a vector space with dimension 𝑑. To choose an orthonormal basis {𝑒 _𝑖 } 𝑖=0,...,𝑑−1

of 𝑉, the following equation holds in the Clifford algebra 𝐶ℓ(𝑉):

{𝑒 _𝑖 , 𝑒 _𝑗 } = 𝑒 _𝑖 𝑒 _𝑗 + 𝑒 _𝑗 𝑒 _𝑖 = 2𝜂 _𝑖𝑗 , (IV.1) where 𝜂 _𝑖𝑗 = diag(−1, 1, 1, ..., 1). The Clifford algebra of the 𝑑 dimensional vec- tor space has 2 ^𝑑 dimensions, and the basis can be expressed by {𝑒 _𝑖 } 𝑖=0,...,𝑑−1

as

1, 𝑒 _𝑖 , 1

2 𝑒 _[𝑖 𝑒 _𝑗] , 1

3! 𝑒 _[𝑖 𝑒 _𝑗 𝑒 _𝑘] , … , 𝑒 ₁ 𝑒 ₂ ⋯ 𝑒 _𝑑 , (IV.2) where the square brackets in the indices mean the antisymmetric permutation.

Remarkably, there is a Lie algebra representation 𝔰𝔬(𝑛, 𝑑 − 𝑛) → 𝐶ℓ(𝑉) such

16

CHAPTER IV. TYPE IIA THEORY OF SUPERGRAVITY 17 that

𝑀 _𝑖𝑗 ↦ 1

2 [𝑒 _𝑖 , 𝑒 _𝑗 ] = 𝑒 _𝑖 𝑒 _𝑗 − 𝑒 _𝑗 𝑒 _𝑖 , (IV.3) where 𝑀 _𝑖𝑗 is a basis of 𝔰𝔬(1, 𝑑 − 1). However, there is a duplication of 𝑒 _𝑖 ↔ −𝑒 _𝑖 . Therefore ¹ ₂ [𝑒 _𝑖 , 𝑒 _𝑗 ] spans 𝑆𝑝𝑖𝑛(1, 𝑑−1) which is the double cover of 𝑆𝑂(1, 𝑑−1).

A matrix representation of the Clifford algebra 𝐶ℓ(𝑉) → 𝐺𝐿(2𝑑+2); 𝑒 _𝑖 ↦ Γ ^𝑖 is called gamma matrices (we consider the even dimensional case here only).

This representation is constructed recursively. the 𝐷 = 2𝑘 + 2 dimensional gamma matrices Γ ^𝜇 is given by the 2𝑘 = 2(𝑘 − 1) + 2 dimensional gamma matrix 𝛾 ^𝑖 as follows:

Γ ^𝑖 = 𝛾 ^𝑖 ⊗ ⎛ ⎜

⎝ 1 0 0 −1

⎞ ⎟

⎠

, (IV.4)

Γ ^𝐷−2 = 𝟏 ⊗ ⎛ ⎜

⎝ 0 1 1 0

⎞ ⎟

⎠

, (IV.5)

Γ ^𝐷−1 = 𝟏 ⊗ ⎛ ⎜

⎝ 0 −i

i 0

⎞ ⎟

⎠

, (IV.6)

where 𝟏 is the 2 ^𝑘 × 2 ^𝑘 identity matrix. In 𝑘 = 1 case, the gamma matrices are given by Pauli matrices,

⎛ ⎜

⎝ 0 1 1 0

⎞ ⎟

⎠ , ⎛ ⎜

⎝ 0 −i

i 0

⎞ ⎟

⎠ , ⎛ ⎜

⎝ 1 0 0 −1

⎞ ⎟

⎠ . (IV.7)

𝐷 = 2𝑘 + 2 dimensional spacetime has 2 ^𝑘+1 × 2 ^𝑘+1 gamma matrices and the (Majorana) spinors with 2 ^𝑘+1 components as the vector on which the Clifford algebra acts.

IV.2 Supersymmetry

Supersymmetry has been introduced in the context of the gauge unification of quantum interactions. Coleman–Mandula theorem states that the S-matrix can have the symmetry that is only the direct product of the Poincaré algebra and the (non-graded) Lie algebra of the internal symmetry. Supersymmetry is graded, so it is beyond the assumption of this statement.

Haag–Łopuszański–Sohnius theorem defines the graded algebra as below, [𝑀 ^𝜇𝜈 , 𝑀 ^𝜌𝜎 ] = 𝑖(𝜂 ^𝜇𝜌 𝑀 ^𝜈𝜎 − 𝜂 ^𝜇𝜎 𝑀 ^𝜈𝜌 − 𝜂 ^𝜈𝜌 𝑀 ^𝜇𝜎 + 𝜂 ^𝜈𝜎 𝑀 ^𝜇𝜌 ) , (IV.8) {𝑄 ^𝑖 , ̄ 𝑄 ^𝑗 } = 𝛿 ^𝑖𝑗 𝛾 _𝜇 𝑃 ^𝜇 + 𝟏𝑆 ^𝑖𝑗 + 𝑖𝛾 ₅ 𝑉 ^𝑖𝑗 , (IV.9) [𝑄 ^𝑖 _𝛼 , 𝑀 _𝜇𝜈 ] = 1

2 (𝜎 _𝜇𝜈 ) _𝛼𝛽 )𝑄 ^𝑖 _𝛽 , (IV.10)

where 𝜎 _𝜇𝜈 = [Γ _𝜇 , Γ _𝜈 ], 𝑃 _𝜇 and 𝑀 _𝜇𝜈 are the Poincaré generators, 𝑄 ^𝑖 _𝛼 are the

fermionic supercharges, and 𝑆 ^𝑖𝑗 and 𝑉 ^𝑖𝑗 are central charges. And the algebra

with the internal Lie group and commutative one are abbreviated. Central

charges are commutative with the other generators. When the index 𝑖 runs

over 1, 2, ..., 𝑁, the algebra above is called 𝒩 = 𝑁 extended supersymmetry.

CHAPTER IV. TYPE IIA THEORY OF SUPERGRAVITY 18 In the massless case, putting the frame 𝑃 ^𝜇 = (|𝑝|, 0, 0, |𝑝|), we can construct the creation-annihilation operators by

𝑆 ^𝑖 = 1

4|𝑝| (𝑄 ^𝑖 _𝑖 + i𝑄 ^𝑖 ₄ ) , (IV.11) and its †-conjugate. The helicity 𝜆 − 𝑘/2 state in the supersymmetric multiplet with maximal helicity 𝜆 is obtained by the helicity 𝜆 state |𝜆⟩ and the creation- annihilation operators as follows,

𝑆 _𝑖

𝑆 _𝑖

⋯ 𝑆 _𝑖

|𝜆⟩ . (IV.12) The helicity 𝜆 − 𝑘/2 state has ⎛ ⎜

⎝ 𝑁

𝑘

⎞ ⎟

⎠

modes where ⎛ ⎜

⎝ 𝑛 𝑘

⎞ ⎟

⎠

= 𝑛!

(𝑛 − 𝑘)! 𝑘! are binomial coefficients.

By the Weinberg–Witten theorem, we think that the maximal spin is 2. The spin-2 particle is a graviton. And the maximal supersymmetry allowing spin-2 particles is 𝒩 = 8 case. In the 4 dimensional spacetime, the number of the com- ponents of Majorana spinor is 4. So there are 32 supercharges. On the other hand, the Majorana spinor in eleven spacetime has 32 components. Therefore, the maximally supersymmetric case gives eleven dimensional supergravity.

IV.3 Supergravity

The eleven dimensional supergravity theory has 256 =

8

∑

𝑘=0

⎛ ⎜

⎝ 8 𝑘

⎞ ⎟

⎠

(IV.13) d.o.f. (degrees of freedom) and it is divided into 128 bosonic and 128 fermionic parts. In the eleven dimensional space, the graviton 𝑔 _𝑀𝑁 with ^12×11 ₂ −11−11 = 44 d.o.f. and antisymmetric 3-form field 𝐶 ₃ = ∑ _{𝐿,𝑀,𝑁 3!} ¹ 𝐶 _𝐿𝑀𝑁 d𝑥 ^𝐿 ∧d𝑥 ^𝑀 ∧d𝑥 ^𝑁 with ₉ 𝐶 ₃ = ^9×8×7 _3×2×1 = 84 satisfy this d.o.f. Now, the diffeomorphism invariance and the Bianchi identity reduce the d.o.f. of graviton by −11 − 11, respectively.

The action functional for them without higher derivative terms is decided by the symmetry as follows:

𝑆 ₁₁ = ∫ (𝑅 ⋆ 1 − 1

2 𝐹 ₄ ∧ ⋆𝐹 ₄ ) − 1

12𝜅 ² ₁₁ ∫ 𝐴 ₃ ∧ 𝐹 ₄ ∧ 𝐹 ₄ . (IV.14) Type IIA supergravity is obtained by the 𝑆 ¹ compactification from this eleven dimensional supergravity, as mentioned before. The eleven dimensional sin- gle supercharge with 32 components is divided into two supercharges with 16 components which are Majorana spinors in ten dimensional spacetime. The bosonic fields 𝑔 _𝑀𝑁 and 𝐶 _𝐿𝑀𝑁 are decomposed into 𝜙, 𝐴 _𝑚 , 𝑔 _𝑚𝑛 , 𝐵 _𝑚𝑛 and 𝐶 _ℓ𝑚𝑛 as

1 2

10

∑

𝑀,𝑁=0

𝑔 _𝑀𝑁 d𝑥 ^𝑀 ⊙ d𝑥 ^𝑁 =

𝜙 d𝑥 ¹⁰ ⊙ d𝑥 ¹⁰ +

9

∑

𝑚=0

𝐴 _𝑚 d𝑥 ^𝑚 ⊙ d𝑥 ¹⁰ + 1 2

9

∑

𝑚,𝑛=0

𝑔 _𝑚𝑛 d𝑥 ^𝑚 ⊙ d𝑥 ^𝑛 ,

(IV.15)

CHAPTER IV. TYPE IIA THEORY OF SUPERGRAVITY 19

𝐶 ₃ = 1 2

9

∑

𝑚,𝑛=0

𝐵 _𝑚𝑛 d𝑥 ^𝑚 ∧ d𝑥 ^𝑛 ∧ d𝑥 ¹⁰ + 1 3!

9

∑

ℓ,𝑚,𝑛=0

𝐶 _ℓ𝑚𝑛 d𝑥 ^ℓ ∧ d𝑥 ^𝑚 ∧ d𝑥 ^𝑛 , (IV.16) where ⊙ is symmetric and ∧ is anti-symmetric tensor product.

The action functional is fixed by supersymmetry as follows:

𝑆 ₁₀ = 1

2 ∫ [ 𝑒 ^−2𝜙 (𝑅 ⋆ 1 + 4d𝜙 ∧ ⋆d𝜙 − 1

2 𝐻 ₃ ∧ ⋆𝐻 ₃ )

− 𝐹 ₂ ∧ ⋆𝐹 ₂ − 𝐹 ₄ ∧ ⋆𝐹 ₄ − 𝐵 ₂ ∧ d𝐶 ₃ ∧ d𝐶 ₃ ] ,

(IV.17)

where 𝐻 ₃ ≡ d𝐵 ₂ , 𝐹 ₂ ≡ d𝐶 ₁ , and 𝐹 ₄ ≡ d𝐶 ₃ − 𝐶 ₁ ∧ 𝐻 ₃ . Now the symbol 𝐶 ₃ is redefined to indicate the second term of IV.16.

However, it is just a introduction of type IIA supergravity. The more fun-

damental theory of quantum gravity is given by the superstring theory, that is

introduced in next chapter.

V

Type IIA Theory of Strings

V.1 Free Superstring

String theory considers relativistic strings and branes flying in general dimen- sional space interacting with each others. Specifically, we consider the type IIA superstrings. This theory has 𝒩 = 2 supersymmetric string spectra in 10 dimensional spacetime. Such a string is described by world sheet theory,

𝑆 = − 1

𝜋 ∫ d𝜎d𝜏√− det Π _𝛼 ⋅ Π _𝛽 + ∫ Ω ₂ , (V.1) where 𝜏 is a proper time for the string similar to the ordinary relativistic par- ticle theory. 𝜎 parametrizes the string itself. And

Π ^𝜇 _𝛼 = 𝜕 _𝛼 𝑋 ^𝜇 − ∑

𝐴

̄ Θ ^𝐴 Γ ^𝜇 𝜕 _𝛼 Θ ^𝐴 , (V.2) where 𝑋 ^𝜇 indicates the position of the string in the spacetime and Θ represents the corresponded supersymmetric modes which are Graßmanian coordinates.

We consider the type IIA theory i.e. 10 dimensional spacetime and 𝒩 = 2 supersymmetry, 𝐴 = 1, 2 and Θ is Majorana–Weyl spinor whose components are 16. Θ satisfies

Γ ₁₁ Θ ^𝐴 = (−1) ^𝐴+1 Θ ^𝐴 , (V.3) in our type IIA case where Γ ₁₁ = ∏ ⁹ _𝑘=0 Γ _𝑘 . The action 𝑆 has the spacetime super-Poincaré symmetry, worldsheet diffeo-morphism invariance, and 𝜅-symmetry which is conserved due to the topological term ∫ Ω ₂ . The last symmetry elim- inates half of the fermionic degree of freedom. To fix the coordinate as

𝑋 ⁺ = 𝑥 ⁺ + ℓ ² _𝑠 𝑝 ⁺ 𝜏 , (V.4) a.k.a. light-cone coordinate where 𝑋 ⁺ = ¹

√2 (𝑋 ⁰ + 𝑋 ⁹ ) and 𝑥 ⁺ and 𝑝 ⁺ are the string position and the momentum respectively, the remaining 8 transverse modes 𝑋 ¹ , ...𝑋 ⁸ expresses the physical d.o.f. It gives massless vector represen- tation 𝟖 _𝐯 which belongs to the representation of the little group 𝑆𝑂(8) in the Poincaré group. And putting

Γ ⁺ Θ ^𝐴 = 0 , (V.5)

20

CHAPTER V. TYPE IIA THEORY OF STRINGS 21 Table V.1: Bosonic fields in type IIA theory

NS-NS sector R-R sector dilaton 𝜙 1-form field 𝐶 ₁ 2-form field 𝐵 ₂ 3-form field 𝐶 ₃ metric tensor 𝑔 _𝑀𝑁

where Γ ⁺ = ¹

√2 (Γ ⁰ + Γ ⁹ ), the fermionic physical d.o.f. is estimated by 2 × 8 due to the 𝜅-symmetry. They give the spinor representations 𝟖 _𝐬 and 𝟖 _𝐜 which belong to the little group 𝑆𝑝𝑖𝑛(8). They represents the undotted and dotted spinor, respectively.

The free string massless spectrum can be derived. In the type IIA theory, there are representations in the bosonic part:

𝟖 _𝐯 ⊗ 𝟖 _𝐯 = 𝟏 + 𝟐𝟖 + 𝟑𝟓 , (V.6)

𝟖 _𝐬 ⊗ 𝟖 _𝐜 = 𝟖 _𝐯 + 𝟓𝟔 _𝐭 . (V.7)

The former representation is called NS–NS sector and the latter is called R–R sector.

Their spectra become physical quantum fields in a point particle limit ℓ _𝑠 → 0 where ℓ _𝑠 is a string length scale given by ℓ ² _𝑠 /2 = 𝛼 ^′ . In this limit, the spectra corresponds to the fields in the table.V.1.

And it is known that the dynamics of these fields is governed by following action (dictated by the symmetries);

𝑆 = 1

2 ∫ [𝑒 ^−2𝜙 (𝑅 ⋆ 1 + 4d𝜙 ∧ ⋆d𝜙 − 1

2 𝐻 ₃ ∧ ⋆𝐻 ₃ )

−𝐹 ₂ ∧ ⋆𝐹 ₂ − 𝐹 ₄ ∧ ⋆𝐹 ₄ − 𝐵 ₂ ∧ d𝐶 ₃ ∧ d𝐶 ₃ + 𝑂(𝛼 ^′ )] ,

(V.8)

where 𝐻 ₃ , 𝐹 ₂ , and 𝐹 ₄ are NS/R-R flux field strengths defined by

𝐻 ₃ ≡ d𝐵 ₂ , 𝐹 ₂ ≡ d𝐶 ₁ , and 𝐹 ₄ ≡ d𝐶 ₃ − 𝐶 ₁ ∧ 𝐻 ₃ . (V.9) It is just a bosonic part of type IIA supergravity action except the higher order terms 𝑂(𝛼 ^′ ). This means the type IIA supergravity is the low energy effective theory for the massless modes of the type IIA superstring theory. The higher order terms of the string scale 𝑂(𝛼 ^′ ) are called 𝛼 ^′ -corrections.

V.2 Interacting Strings

The above discussion in V.1 is devoted to free single string. However, strings

interact with each other. The worldsheet composed by interacting 𝑘 closed

strings is regarded as the Riemannian surface with 𝑘 punctures. Asymptotic

string states correspond to vertex operators. And the vertex operators are at the

punctures on the Riemannian surface. Riemannian sphere gives a tree-level

amplitude. 1-loop amplitude corresponds to genus one Riemannian surface

a.k.a. a torus. 𝑔 _s = ⟨𝑒 ^𝜙 ⟩, the vacuum expection value of the dilation 𝜙 gives the

parameter of this loop ‘expansion’. Therefore, the loop-corrections are called

𝑔 _s -corrections.

CHAPTER V. TYPE IIA THEORY OF STRINGS 22

V.3 Branes

Branes, the other important actors in string theories, are 𝑝-dimensional objects and have (𝑝 + 1)-dimensional worldvolume. They interact and generate strings.

There are two types of branes which are called D𝑝-branes and NS𝑝-branes, respectively.

Historically these branes were introduced in two faces as follows:

• (in D-brane) as an object with which open strings can attach their ends, as Dirichlet boundary conditions, and

• (in both D- and NS- brane) as stringy BPS-‘solitons’ with RR or NS charge.

The first face arises in the literature of T-duality. When we consider the space with ℛ ^𝑑 × 𝑆 ¹ , a T-duality transformation makes the radius of 𝑆 ¹ inverse. This transformation replaces the Neumann boundary conditions by the Dirichlet boundary conditions. Therefore, free open string theory needs the objects with which open strings can attach their ends (see textbook [BBS06], for example).

Second, branes are introduced as the objects which interact with 𝑝-form string fields as follows:

𝑒 _NS ∫

𝒱

𝐵 ₂ , 𝑒 ₀ ∫

𝒱

𝐶 ₁ , and 𝑒 ₂ ∫

𝒱

𝐶 ₃ , (V.10) like an electromagnetic field 𝐴 interacts with a point particle by

𝑒 ∫ 𝐴 = 𝑒 ∫ 𝐴 _𝜇 (d𝑥 ^𝜇 /d𝜏)d𝜏 . (V.11) The integrals are over the brane world volumes 𝒱 _∗ , and 𝑒 _∗ is an electric charge.

A brane which interacts with 𝐵 ₂ is called NS5-brane. The others are called D𝑝-branes. D𝑝-brane interacts with (𝑝 + 1)-form field.

Type IIA theory has D0-, D2-, D4-, and D6-branes. D0- and D2- branes interact electrically with 𝐶 ₁ and 𝐶 ₃ , respectively. And D4- and D6- branes have their magnetic dual. Their charges are obtained by following Gaußian integrals;

𝑒 _𝑝 = ∫

𝑆

⋆𝐹 _𝑝+1 , 𝑚 _𝑝 = ∫

𝑆

𝐹 _𝑝+1 , (V.12) where 𝑆 ^𝑝+2 is a (𝑝 + 2)-dimensional sphere surrounding the D𝑝-brane. They satisfy the Dirac quantization condition 𝑒 _𝑝 𝑚 _6−𝑝 ∈ ℤ.

A D-brane has its charge and mass which satisfy the BPS bound. Due to the 𝒩 = 2 supersymmetry with the central charge 𝑍 ^𝑖𝑗 = 𝜖 ^𝑖𝑗 𝑍, the supersymmetry algebra in the rest frame

{𝑄 ^𝑖 _𝛼 , ̄ 𝑄 ^𝑗 _{𝛽 ̇} } = 2𝑀𝛿 _{𝛼 ̇ 𝛽} 𝛿 ^𝑖𝑗 , (V.13) {𝑄 ^𝑖 _𝛼 , 𝑄 ^𝑗 _𝛽 } = 2𝑍𝜖 _𝛼𝛽 𝜖 ^𝑖𝑗 , (V.14) { ̄ 𝑄 ^𝑖 _{𝛼 ̇} , ̄ 𝑄 ^𝑗 _{𝛽 ̇} } = 2𝑍𝜖 _{𝛼 ̇ ̇ 𝛽} 𝜖 ^𝑖𝑗 , (V.15) gives the bound [Moo]

𝑀 ≥ 𝑍 . (V.16)

The BPS bound guarantees the stability of D-brane. And stable D-branes

contribute to non-perturbative corrections (in 𝑔 _s ) to be studied below.

VI

Calabi–Yau Compactification

This chapter is devoted to the background geometry and its effect to type IIA supergravity i.e. the low energy effective theory of type IIA superstrings.

We consider a direct product 𝑀 × 𝔜 as the background spacetime where 𝑀 is ordinary four dimensional Minkowski spacetime and 𝔜 is Calabi–Yau 3-fold with six real dimensions. This background setting is called Calabi–Yau compact- ification. We think the our universe begins with this background space 𝑀 × 𝔜, at least in the inflationary age. As mentioned before, Calabi–Yau compactifi- cation gives 𝒩 = 2 supersymmetry. This symmetry guarantees, in principle, exact calculation of quantum corrections in the theory. It is a motivation to use Calabi–Yau compactification in this dissertation.

VI.1 Definition of Calabi–Yau manifold

Calabi–Yau 𝑛-fold 𝔜 is defined by the holonomy group which is subgroup of 𝑆𝑈(𝑛). A holonomy group indicates the result of a parallel transport of a tangent vector along a closed curve. Calabi–Yau manifold is also a Kähler manifold, but Ricci-flat. Therefore A Calabi–Yau manifold allows complex and Kähler structures. Their structures define the field components of the theory.

An almost complex structure is a tensor 𝒥 which defines how even real components are divided into real and imaginary part. Formally, it is defined by 𝒥 ∶ 𝑇𝔜 → 𝑇𝔜 that 𝒥 ∘ 𝒥 = −𝟏 where 𝟏 is the identity matrix with an appropriate dimension. In component-wise form, there are coordinates (𝑥 ¹ , 𝑥 ² , … , 𝑥 ^𝑛 ; 𝑦 ¹ , 𝑦 ² , … , 𝑦 ^𝑛 ) that,

𝑛

∑

𝑚=1

𝒥 _ℓ ^𝑚 𝜕

𝜕𝑥 ^𝑚 = 𝜕

𝜕𝑦 ^ℓ ,

𝑛

∑

𝑚=1

𝒥 _ℓ ^𝑚 𝜕

𝜕𝑦 ^𝑚 = − 𝜕

𝜕𝑥 ^ℓ . (VI.1) When the Nijenhuis tensor for the almost complex structure 𝒥 vanishes,

𝑁 _𝒥 (𝑋, 𝑌) = [𝒥𝑋, 𝒥𝑌] − 𝒥[𝒥𝑋, 𝑌] − 𝐽[𝑋, 𝒥𝑌] − [𝑋, 𝑌] = 0 , (VI.2) 𝒥 is integrable and called a complex structure.

23

CHAPTER VI. CALABI–YAU COMPACTIFICATION 24 A Kähler structure is based on the symplectic structure 𝜔. 𝜔 is a non-zero two-form field which satisfies d𝜔 = 0. There are coordinates in which

𝜔 =

𝑛

∑

𝑚=1

d𝑞 ^𝑚 ∧ d𝑝 _𝑚 . (VI.3)

(𝑞 ¹ , 𝑞 ² , … , 𝑞 ^𝑛 ; 𝑝 ₁ , 𝑝 ₂ , … , 𝑝 _𝑛 ) are the pairs of the positions and the momenta in the context of the Hamiltonian dynamics. Therefore, the symplectic form de- fines how even real components are divided into the position and the momen- tum parts.

And these two structures, complex and symplectic, and the Riemannian structure (a metric 𝑔) are compatible when,

𝑔(𝑉, 𝑊) = 𝑔(𝒥𝑉, 𝒥𝑊) , (VI.4)

𝜔(𝑉, 𝑊) = 𝑔(𝒥𝑉, 𝑊) , (VI.5)

where 𝜔 is called Kähler form. Let us denote Kähler form by 𝐽. 𝐽 can be ex- pressed by

𝐽 = i𝑔 _{𝑖 ̄𝚥} d𝑧 ^𝑖 d ̄𝑧 ^̄𝚥 . (VI.6) A Calabi–Yau manifold has these compatible structures also. The manifold with this Kähler form is called Kähler manifold. Kähler manifold is also defined when its holonomy group is a subgroup of 𝑈(𝑛).

Now, Kähler manifold has Kähler potential 𝒦 such that

𝑔 _{𝑖 ̄𝚥} = 𝜕 _𝑖 𝜕 𝒦 ≡ 𝒦 _{̄𝚥 𝑖 ̄𝚥} , (VI.7) and Kähler prepotential which gives Kähler potential as

𝒦 = − log [i ( ̄ 𝑋 ^𝐼 𝐹 _𝐼 − 𝑋 ^𝐼 𝐹 ̄ _𝐼 )] = − log [−2ℑ ( ̄ 𝑋 ^𝐼 𝐹 _𝐼 )] , (VI.8) where 𝑋 ^𝐼 are homogeneous coordinates on the manifold and 𝐹 _𝐼 = 𝜕 _𝑋

𝐹.

When Kähler manifold has a flat and torsion-free connection ∇ which sat- isfies

(d _∇ 𝒥)(𝑋, 𝑌) = 0 , (VI.9)

∇𝜔 = 0 , (VI.10)

it is called special Kähler manifold.

Moreover, these structures can deform. Therefore, we consider the continu- ous sequence of deformations of Calabi–Yau 3-fold in our Minkowski spacetime.

This means we treat a fibration 𝑀 × ℳ _𝔜 → 𝑀 where ℳ _𝔜 is a moduli space of Calabi–Yau manifold. A section of this fibration gives new fields. Of course, ℳ _𝔜 is factorized into two subspaces; Kähler moduli space 𝒦 _𝐾 (𝔜) and complex moduli space 𝒦 _𝐶 (𝔜). They become special Kähler manifolds. Their infinitesi- mal deformations are given by Kodaira–Spencer theory in Dolbeaut cohomology with coefficient 𝐾 in 𝐻 ^𝑝,𝑞 (𝔜, 𝐾). Complex deformations are obtained by

𝜒 = Ω _𝑖𝑗𝑘 𝛿𝒥 ^𝑘 _̄ℓ d𝑧 ^𝑖 ∧ d𝑧 ^𝑗 ∧ d𝑧 ^̄ℓ ∈ 𝐻 ^2,1 (𝔜, 𝐂) , (VI.11) and Kähler deformations are obtained by

𝛿𝐽 = i𝛿𝑔 _{𝑖 ̄𝚥} d𝑧 ^𝑖 d ̄𝑧 ^̄𝚥 ∈ 𝐻 ^1,1 (𝔜, 𝐑) , (VI.12)

CHAPTER VI. CALABI–YAU COMPACTIFICATION 25

ℎ ^2,1 ℎ ^1,1 ℎ ^2,1

ℎ ^1,1

1 1

1

1

0 0 0

0

0 0 0

0

Figure VI.1: Hodge diamond for Calabi–Yau threefolds

where Ω is the holomorphic (3,0)-form belonging to 𝐻 ^3,0 (𝔜). The coordinates of the space 𝒦 _𝐶 (𝔜) and 𝒦 _𝐾 (𝔜) are called moduli fields, or moduli simply.

The number of these moduli is counted by the dimension of the cohomol- ogy class. For the Calabi–Yau 3-form we use, it is drawn by the diagram known as the Hodge diamond in fig.VI.1.

A Calabi–Yau manifold 𝔜 has cycles which belong to homology class 𝐻 _∗ (𝔜) of 𝔜 dual to cohomology. Different cycle give different independent physical modes. With their modes, complex moduli become components of hypermul- tiplets, and Kähler moduli become components of 𝒩 = 2 vector multiplets.

VI.2 Field Content

Now, we define the notation of homology and cohomology basis by the ta- ble.VI.1, where 𝒜 _Λ and ℬ ^Σ are chosen to make the intersection numbers

⟨𝒜 _Λ , ℬ ^Σ ⟩ = ∫

𝔜 𝛽 _Λ ∧ 𝛼 ^Σ = 𝛿 ^Σ _Λ . (VI.13) And 𝜔 _𝑖 and 𝜔 ^𝑗 satisfy

𝜔 _𝑖 ∧ 𝜔 ^𝑗 = 𝛿 _𝑖 ^𝑗 𝜔 _𝔜 , 𝜔 _𝑖 ∧ 𝜔 _𝑗 = 𝜅 _𝑖𝑗𝑘 𝜔 _𝑘 , (VI.14) where 𝜔 _𝔜 is the volume form spanning 𝐻 ⁶ (𝔜) and 𝜅 _𝑖𝑗𝑘 are triple intersection numbers defined by

𝜅 _𝑖𝑗𝑘 = ∫

𝔜 𝜔 _𝑖 ∧ 𝜔 _𝑗 ∧ 𝜔 _𝑘 = ⟨𝛾 _𝑖 , 𝛾 _𝑗 , 𝛾 _𝑘 ⟩ . (VI.15) And let us denote the coordinates of complex moduli space by

𝑋 ^Λ = [𝑋 ⁰ ∶ 𝑋 ¹ ∶ … ∶ 𝑋 ^ℎ

] = [1∶ 𝑢 ¹ ∶ … ∶ 𝑢 ^ℎ

] = [1∶ 𝑢 ^𝑎 ] , (VI.16) with suffix 𝑎 running over 1, ..., ℎ ^2,1 , and

𝑧 ^𝑖 = 𝑏 ^𝑖 + i𝑡 ^𝑖 = ∫

𝛾

(𝐵 ₂ + i𝐽) , (VI.17)

CHAPTER VI. CALABI–YAU COMPACTIFICATION 26 Table VI.1: Notation of homology and cohomology basis of Calabi–Yau three- folds.

set basis dual basis range of suffix 𝐻 ₃ (𝔜) 𝒜 _Λ ; ℬ ^Σ 𝛽 _Λ ; 𝛼 ^Σ Λ = 0, 1, ..., ℎ ^2,1 𝐻 ₂ (𝔜), 𝐻 ₄ (𝔜) 𝛾 ^𝑖 , 𝛾 _𝑖 𝜔 _𝑖 , 𝜔 ^𝑖 𝑖 = 1, ..., ℎ ^1,1

Table VI.2: 𝑁 = 2 supersymmetry multiplets of the fields on the tableVI.3.

# type of multiplet bosonic fields 1 gravitational 𝑔 _𝜇𝜈 , 𝐶 ⁰ ₁ = 𝐴 _𝜇 d𝑥 ^𝜇 1 tensor 𝐵 ₂ , 𝜙, 𝜁 ⁰ , ̃ 𝜁 ₀ ℎ ^2,1 hyper- 𝑢 ^𝑎 , 𝜁 ^𝑎 , ̃ 𝜁 _𝑎 ℎ ^1,1 𝒩 = 2 vector- 𝐶 ₁ ^𝑖 , 𝑧 ^𝑖

1 universal hyper- 𝜎, 𝜙, 𝜁 ⁰ , ̃ 𝜁 ₀

where 𝐵 ₂ is 2-form field in NS-NS sector, 𝐽 is Kähler form defined above. 𝑋 ^𝐼 also means [1∶ 𝑧 ^𝑖 ]. Using them, field content in type IIA supergravity on Calabi–

Yau compactification can be enumerated by the table VI.3.

And it composes 𝒩 = 2 supermultiplets listed in the table VI.2.

Now we remark about ‘4d dual’ given here. In general, 𝑝-form fields have their Poincaré dual partners[LM02] because, in massless case, 𝑝-form field in 𝑑 dimensional spacetime has ⎛ ⎜

⎝ 𝑑 − 2

𝑝

⎞ ⎟

⎠

physical degrees of freedom. And its Poincaré dual (𝑑 − 𝑝 − 2)-form field has same d.o.f. In the massive case 𝑝-form field is dualized by (𝑑 − 𝑝 − 1)-form field.

Let us denote the ‘4d dual’ of 𝐵 ₂ and 𝐶 ₃ form fields by 𝜎 and 𝑒 ₀ respec- tively. For instance, the generic action for 2-form field and its dual field is given through the relation ⋆𝐻 ₃ = ¹ _𝑔 (d𝜎 + 𝐽 ₁ ) by

𝑆 _𝐵

= − ∫ [ 𝑔

4 𝐻 ₃ ∧ ⋆𝐻 ₃ − 1

2 𝐻 ₃ ∧ 𝐽 ₁ ] , (VI.18) 𝑆 _𝜎 = − ∫ 1

4𝑔 (d𝜎 + 𝐽 ₁ ) ∧ ⋆(d𝜎 + 𝐽 ₁ ) . (VI.19) A 3-form field in 4 dimensional space has no physical d.o.f. Therefore, its dual

‘field’ 𝑒 ₀ is constant. They are related by 𝑔

2 ⋆ (d𝐶 ₃ − 𝐽 ₄ ) = − ℎ + 𝑒 ₀

2 , (VI.20)

and their generic actions are 𝑆 _𝐶

= − ∫ [ 𝑔

4 (d𝐶 ₃ − 𝐽 ₄ ) ∧ ⋆(d𝐶 ₃ − 𝐽 ₄ ) + ℎ

2 d𝐶 ₃ ] , (VI.21) 𝑆 _𝑒

= − ∫ [ 1

4𝑔 (ℎ + 𝑒 ₀ ) ² ⋆ 1 + 1

2 (ℎ + 𝑒 ₀ )𝐽 ₄ ] . (VI.22)

CHAPTER VI. CALABI–YAU COMPACTIFICATION 27 Table VI.3: Field content in type IIA supergravity in Calabi–Yau compactifica- tion. This table is borrowed from [Ale13] and changed slightly. The suffix 𝑎 runs over 𝑎 = 1, … , ℎ ^1,1 .

NS-NS R-R

10d 𝑔 ̂ _𝑁𝑀 𝐵 ̂ ₂ 𝜙 ̂ 𝐶 ̂ ₁ 𝐶 ₃ ̂

4d 𝑔 _𝜇𝜈 𝑡 ^𝑖 𝑢 ^𝑎 𝐵 ₂ + 𝑏 ^𝑖 𝜔 _𝑖 𝜙 𝐶 ₁ ⁰ 𝐶 ₃ + 𝐶 ^𝑖 ₁ ∧ 𝜔 _𝑖 + 𝜁 ^Λ 𝛼 _Λ + ̃ 𝜁 _Σ 𝛽 ^Σ 4d dual 𝑔 _𝜇𝜈 𝑡 ^𝑖 𝑢 ^𝑎 𝜎 𝑏 ^𝑖 𝜙 𝐶 ₁ ⁰ 𝑒 ₀ 𝐶 ^𝑖 ₁ 𝜁 ^Λ 𝜁 _Σ ̃

Now 𝐽 _1,4 are generic forms coupled with 𝑝-form fields.

Actually the last line of the table VI.2 indicates a hypermultiplet 4d dual to tensor multiplet. This multiplet always appears even when ℎ ^2,1 vanishes.

Therefore, it is called universal and plays important role in our discussion. A Calabi–Yau manifold with no complex moduli i.e. ℎ ^2,1 = 0 is called rigid. Rigid Calabi–Yau compactification has only one hypermultiplet i.e. the universal one. 𝜎 is called NS-axion because it has a Heisenberg shift symmetry discussed below.

The dynamics of moduli fields is governed by the metric on their moduli space. For complex moduli space, which is special Kähler manifold, the Kähler potential 𝒦 is given by

𝐾 = −i log ∫

𝔜 Ω ∧ ̄ Ω . (VI.23)

Using the Riemannian bilinear relation,

∫ 𝔜 𝜒 ∧ 𝜓 = ∑

Λ

(∫ 𝒜

𝜒 ∫

ℬ

𝜓 − ∫

𝒜

𝜓 ∫

ℬ

𝜒) , (VI.24) the prepotential is obtained by 𝐹(𝑋) = ¹ ₂ 𝑋 ^Λ 𝐹 _Λ where

𝑋 ^Λ = ∫

𝒜

Ω , 𝐹 _Λ = ∫

ℬ

Ω . (VI.25)

For Kähler moduli space, which is also special Kähler, the metric is intro- duced naturally by

𝑔 _{𝑖 ̄𝚥} = 1 𝒱 ∫

𝔜 𝜔 _𝑖 ∧ ⋆𝜔 _𝑗 = 𝜕 _𝑖 𝜕 _̄𝚥 (− log 8𝒱) , (VI.26) where 𝒱 is Calabi–Yau volume

𝒱 = 1 6 ∫

𝔜 𝐽 ∧ 𝐽 ∧ 𝐽 = 1

6 𝜅 _𝑖𝑗𝑘 𝑡 ^𝑖 𝑡 ^𝑗 𝑡 ^𝑘 . (VI.27) The prepotential is given by

𝐹(𝑋) = − 1 6

𝜅 _𝑖𝑗𝑘 𝑋 ^𝑖 𝑋 ^𝑗 𝑋 ^𝑘

𝑋 ⁰ . (VI.28)